Theorem ringabl

Description: A ring is an Abelian group. (Contributed by NM, 26-Aug-2011)

	Ref	Expression
Assertion	ringabl	|- ( R e. Ring -> R e. Abel )

Proof

Step	Hyp	Ref	Expression
1		eqidd	|- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
2		eqidd	|- ( R e. Ring -> ( +g ` R ) = ( +g ` R ) )
3		ringgrp	|- ( R e. Ring -> R e. Grp )
4		eqid	|- ( Base ` R ) = ( Base ` R )
5		eqid	|- ( +g ` R ) = ( +g ` R )
6	4 5	ringcom	|- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( +g ` R ) y ) = ( y ( +g ` R ) x ) )
7	1 2 3 6	isabld	|- ( R e. Ring -> R e. Abel )